Theorem coe1tmmul2 22160

Description: Coefficient vector of a polynomial multiplied on the right by a term. (Contributed by Stefan O'Rear, 27-Mar-2015.)

Hypotheses

Ref	Expression
coe1tm.z	⊢ 0 = (0g‘𝑅)
coe1tm.k	⊢ 𝐾 = (Base‘𝑅)
coe1tm.p	⊢ 𝑃 = (Poly1‘𝑅)
coe1tm.x	⊢ 𝑋 = (var1‘𝑅)
coe1tm.m	⊢ · = ( ·𝑠 ‘𝑃)
coe1tm.n	⊢ 𝑁 = (mulGrp‘𝑃)
coe1tm.e	⊢ ↑ = (.g‘𝑁)
coe1tmmul.b	⊢ 𝐵 = (Base‘𝑃)
coe1tmmul.t	⊢ ∙ = (.r‘𝑃)
coe1tmmul.u	⊢ × = (.r‘𝑅)
coe1tmmul.a	⊢ (𝜑 → 𝐴 ∈ 𝐵)
coe1tmmul.r	⊢ (𝜑 → 𝑅 ∈ Ring)
coe1tmmul.c	⊢ (𝜑 → 𝐶 ∈ 𝐾)
coe1tmmul.d	⊢ (𝜑 → 𝐷 ∈ ℕ0)

Assertion

Ref	Expression
coe1tmmul2	⊢ (𝜑 → (coe1‘(𝐴 ∙ (𝐶 · (𝐷 ↑ 𝑋)))) = (𝑥 ∈ ℕ0 ↦ if(𝐷 ≤ 𝑥, (((coe1‘𝐴)‘(𝑥 − 𝐷)) × 𝐶), 0 )))

Distinct variable groups:   𝑥, 0   𝑥,𝐶   𝑥,𝐷   𝑥,𝐾   𝑥, ↑   𝑥,𝐴   𝑥,𝑁   𝑥,𝑃   𝑥,𝑋   𝜑,𝑥   𝑥,𝑅   𝑥, ·   𝑥, ×   𝑥, ∙

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem coe1tmmul2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		coe1tmmul.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ Ring)
2		coe1tmmul.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
3		coe1tmmul.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝐾)
4		coe1tmmul.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ ℕ0)
5		coe1tm.k	. . . . 5 ⊢ 𝐾 = (Base‘𝑅)
6		coe1tm.p	. . . . 5 ⊢ 𝑃 = (Poly1‘𝑅)
7		coe1tm.x	. . . . 5 ⊢ 𝑋 = (var1‘𝑅)
8		coe1tm.m	. . . . 5 ⊢ · = ( ·𝑠 ‘𝑃)
9		coe1tm.n	. . . . 5 ⊢ 𝑁 = (mulGrp‘𝑃)
10		coe1tm.e	. . . . 5 ⊢ ↑ = (.g‘𝑁)
11		coe1tmmul.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑃)
12	5, 6, 7, 8, 9, 10, 11	ply1tmcl 22156	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → (𝐶 · (𝐷 ↑ 𝑋)) ∈ 𝐵)
13	1, 3, 4, 12	syl3anc 1373	. . 3 ⊢ (𝜑 → (𝐶 · (𝐷 ↑ 𝑋)) ∈ 𝐵)
14		coe1tmmul.t	. . . 4 ⊢ ∙ = (.r‘𝑃)
15		coe1tmmul.u	. . . 4 ⊢ × = (.r‘𝑅)
16	6, 14, 15, 11	coe1mul 22154	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ∈ 𝐵 ∧ (𝐶 · (𝐷 ↑ 𝑋)) ∈ 𝐵) → (coe1‘(𝐴 ∙ (𝐶 · (𝐷 ↑ 𝑋)))) = (𝑥 ∈ ℕ0 ↦ (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦)))))))
17	1, 2, 13, 16	syl3anc 1373	. 2 ⊢ (𝜑 → (coe1‘(𝐴 ∙ (𝐶 · (𝐷 ↑ 𝑋)))) = (𝑥 ∈ ℕ0 ↦ (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦)))))))
18		eqeq2 2741	. . . 4 ⊢ ((((coe1‘𝐴)‘(𝑥 − 𝐷)) × 𝐶) = if(𝐷 ≤ 𝑥, (((coe1‘𝐴)‘(𝑥 − 𝐷)) × 𝐶), 0 ) → ((𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))))) = (((coe1‘𝐴)‘(𝑥 − 𝐷)) × 𝐶) ↔ (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))))) = if(𝐷 ≤ 𝑥, (((coe1‘𝐴)‘(𝑥 − 𝐷)) × 𝐶), 0 )))
19		eqeq2 2741	. . . 4 ⊢ ( 0 = if(𝐷 ≤ 𝑥, (((coe1‘𝐴)‘(𝑥 − 𝐷)) × 𝐶), 0 ) → ((𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))))) = 0 ↔ (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))))) = if(𝐷 ≤ 𝑥, (((coe1‘𝐴)‘(𝑥 − 𝐷)) × 𝐶), 0 )))
20		coe1tm.z	. . . . . . 7 ⊢ 0 = (0g‘𝑅)
21	1	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → 𝑅 ∈ Ring)
22		ringmnd 20128	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
23	21, 22	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → 𝑅 ∈ Mnd)
24		ovex 7382	. . . . . . . 8 ⊢ (0...𝑥) ∈ V
25	24	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → (0...𝑥) ∈ V)
26		simprr 772	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → 𝐷 ≤ 𝑥)
27	4	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → 𝐷 ∈ ℕ0)
28		simprl 770	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → 𝑥 ∈ ℕ0)
29		nn0sub 12434	. . . . . . . . . 10 ⊢ ((𝐷 ∈ ℕ0 ∧ 𝑥 ∈ ℕ0) → (𝐷 ≤ 𝑥 ↔ (𝑥 − 𝐷) ∈ ℕ0))
30	27, 28, 29	syl2anc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → (𝐷 ≤ 𝑥 ↔ (𝑥 − 𝐷) ∈ ℕ0))
31	26, 30	mpbid 232	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → (𝑥 − 𝐷) ∈ ℕ0)
32	27	nn0ge0d 12448	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → 0 ≤ 𝐷)
33		nn0re 12393	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ0 → 𝑥 ∈ ℝ)
34	33	ad2antrl 728	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → 𝑥 ∈ ℝ)
35	4	nn0red 12446	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ∈ ℝ)
36	35	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → 𝐷 ∈ ℝ)
37	34, 36	subge02d 11712	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → (0 ≤ 𝐷 ↔ (𝑥 − 𝐷) ≤ 𝑥))
38	32, 37	mpbid 232	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → (𝑥 − 𝐷) ≤ 𝑥)
39		fznn0 13522	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ0 → ((𝑥 − 𝐷) ∈ (0...𝑥) ↔ ((𝑥 − 𝐷) ∈ ℕ0 ∧ (𝑥 − 𝐷) ≤ 𝑥)))
40	39	ad2antrl 728	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → ((𝑥 − 𝐷) ∈ (0...𝑥) ↔ ((𝑥 − 𝐷) ∈ ℕ0 ∧ (𝑥 − 𝐷) ≤ 𝑥)))
41	31, 38, 40	mpbir2and 713	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → (𝑥 − 𝐷) ∈ (0...𝑥))
42	1	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → 𝑅 ∈ Ring)
43		eqid 2729	. . . . . . . . . . . . 13 ⊢ (coe1‘𝐴) = (coe1‘𝐴)
44	43, 11, 6, 5	coe1f 22094	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝐵 → (coe1‘𝐴):ℕ0⟶𝐾)
45	2, 44	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (coe1‘𝐴):ℕ0⟶𝐾)
46	45	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (coe1‘𝐴):ℕ0⟶𝐾)
47		elfznn0 13523	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (0...𝑥) → 𝑦 ∈ ℕ0)
48	47	adantl 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → 𝑦 ∈ ℕ0)
49	46, 48	ffvelcdmd 7019	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → ((coe1‘𝐴)‘𝑦) ∈ 𝐾)
50		eqid 2729	. . . . . . . . . . . . 13 ⊢ (coe1‘(𝐶 · (𝐷 ↑ 𝑋))) = (coe1‘(𝐶 · (𝐷 ↑ 𝑋)))
51	50, 11, 6, 5	coe1f 22094	. . . . . . . . . . . 12 ⊢ ((𝐶 · (𝐷 ↑ 𝑋)) ∈ 𝐵 → (coe1‘(𝐶 · (𝐷 ↑ 𝑋))):ℕ0⟶𝐾)
52	13, 51	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (coe1‘(𝐶 · (𝐷 ↑ 𝑋))):ℕ0⟶𝐾)
53	52	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (coe1‘(𝐶 · (𝐷 ↑ 𝑋))):ℕ0⟶𝐾)
54		fznn0sub 13459	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (0...𝑥) → (𝑥 − 𝑦) ∈ ℕ0)
55	54	adantl 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (𝑥 − 𝑦) ∈ ℕ0)
56	53, 55	ffvelcdmd 7019	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦)) ∈ 𝐾)
57	5, 15	ringcl 20135	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ ((coe1‘𝐴)‘𝑦) ∈ 𝐾 ∧ ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦)) ∈ 𝐾) → (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))) ∈ 𝐾)
58	42, 49, 56, 57	syl3anc 1373	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))) ∈ 𝐾)
59	58	fmpttd 7049	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦)))):(0...𝑥)⟶𝐾)
60	1	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥 − 𝐷)})) → 𝑅 ∈ Ring)
61	3	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥 − 𝐷)})) → 𝐶 ∈ 𝐾)
62	4	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥 − 𝐷)})) → 𝐷 ∈ ℕ0)
63		eldifi 4082	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ((0...𝑥) ∖ {(𝑥 − 𝐷)}) → 𝑦 ∈ (0...𝑥))
64	63, 54	syl 17	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ((0...𝑥) ∖ {(𝑥 − 𝐷)}) → (𝑥 − 𝑦) ∈ ℕ0)
65	64	adantl 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥 − 𝐷)})) → (𝑥 − 𝑦) ∈ ℕ0)
66		eldifsn 4737	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ((0...𝑥) ∖ {(𝑥 − 𝐷)}) ↔ (𝑦 ∈ (0...𝑥) ∧ 𝑦 ≠ (𝑥 − 𝐷)))
67		simplrl 776	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → 𝑥 ∈ ℕ0)
68	67	nn0cnd 12447	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → 𝑥 ∈ ℂ)
69	47	nn0cnd 12447	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ (0...𝑥) → 𝑦 ∈ ℂ)
70	69	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → 𝑦 ∈ ℂ)
71	68, 70	nncand 11480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (𝑥 − (𝑥 − 𝑦)) = 𝑦)
72	71	eqcomd 2735	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → 𝑦 = (𝑥 − (𝑥 − 𝑦)))
73		oveq2 7357	. . . . . . . . . . . . . . . 16 ⊢ (𝐷 = (𝑥 − 𝑦) → (𝑥 − 𝐷) = (𝑥 − (𝑥 − 𝑦)))
74	73	eqeq2d 2740	. . . . . . . . . . . . . . 15 ⊢ (𝐷 = (𝑥 − 𝑦) → (𝑦 = (𝑥 − 𝐷) ↔ 𝑦 = (𝑥 − (𝑥 − 𝑦))))
75	72, 74	syl5ibrcom 247	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (𝐷 = (𝑥 − 𝑦) → 𝑦 = (𝑥 − 𝐷)))
76	75	necon3d 2946	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (𝑦 ≠ (𝑥 − 𝐷) → 𝐷 ≠ (𝑥 − 𝑦)))
77	76	impr 454	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ (𝑦 ∈ (0...𝑥) ∧ 𝑦 ≠ (𝑥 − 𝐷))) → 𝐷 ≠ (𝑥 − 𝑦))
78	66, 77	sylan2b 594	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥 − 𝐷)})) → 𝐷 ≠ (𝑥 − 𝑦))
79	20, 5, 6, 7, 8, 9, 10, 60, 61, 62, 65, 78	coe1tmfv2 22159	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥 − 𝐷)})) → ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦)) = 0 )
80	79	oveq2d 7365	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥 − 𝐷)})) → (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))) = (((coe1‘𝐴)‘𝑦) × 0 ))
81	5, 15, 20	ringrz 20179	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ ((coe1‘𝐴)‘𝑦) ∈ 𝐾) → (((coe1‘𝐴)‘𝑦) × 0 ) = 0 )
82	42, 49, 81	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (((coe1‘𝐴)‘𝑦) × 0 ) = 0 )
83	63, 82	sylan2 593	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥 − 𝐷)})) → (((coe1‘𝐴)‘𝑦) × 0 ) = 0 )
84	80, 83	eqtrd 2764	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥 − 𝐷)})) → (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))) = 0 )
85	84, 25	suppss2 8133	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → ((𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦)))) supp 0 ) ⊆ {(𝑥 − 𝐷)})
86	5, 20, 23, 25, 41, 59, 85	gsumpt 19841	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))))) = ((𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))))‘(𝑥 − 𝐷)))
87		fveq2 6822	. . . . . . . . 9 ⊢ (𝑦 = (𝑥 − 𝐷) → ((coe1‘𝐴)‘𝑦) = ((coe1‘𝐴)‘(𝑥 − 𝐷)))
88		oveq2 7357	. . . . . . . . . 10 ⊢ (𝑦 = (𝑥 − 𝐷) → (𝑥 − 𝑦) = (𝑥 − (𝑥 − 𝐷)))
89	88	fveq2d 6826	. . . . . . . . 9 ⊢ (𝑦 = (𝑥 − 𝐷) → ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦)) = ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − (𝑥 − 𝐷))))
90	87, 89	oveq12d 7367	. . . . . . . 8 ⊢ (𝑦 = (𝑥 − 𝐷) → (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))) = (((coe1‘𝐴)‘(𝑥 − 𝐷)) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − (𝑥 − 𝐷)))))
91		eqid 2729	. . . . . . . 8 ⊢ (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦)))) = (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))))
92		ovex 7382	. . . . . . . 8 ⊢ (((coe1‘𝐴)‘(𝑥 − 𝐷)) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − (𝑥 − 𝐷)))) ∈ V
93	90, 91, 92	fvmpt 6930	. . . . . . 7 ⊢ ((𝑥 − 𝐷) ∈ (0...𝑥) → ((𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))))‘(𝑥 − 𝐷)) = (((coe1‘𝐴)‘(𝑥 − 𝐷)) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − (𝑥 − 𝐷)))))
94	41, 93	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → ((𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))))‘(𝑥 − 𝐷)) = (((coe1‘𝐴)‘(𝑥 − 𝐷)) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − (𝑥 − 𝐷)))))
95	28	nn0cnd 12447	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → 𝑥 ∈ ℂ)
96	27	nn0cnd 12447	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → 𝐷 ∈ ℂ)
97	95, 96	nncand 11480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → (𝑥 − (𝑥 − 𝐷)) = 𝐷)
98	97	fveq2d 6826	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − (𝑥 − 𝐷))) = ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘𝐷))
99	3	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → 𝐶 ∈ 𝐾)
100	20, 5, 6, 7, 8, 9, 10	coe1tmfv1 22158	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘𝐷) = 𝐶)
101	21, 99, 27, 100	syl3anc 1373	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘𝐷) = 𝐶)
102	98, 101	eqtrd 2764	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − (𝑥 − 𝐷))) = 𝐶)
103	102	oveq2d 7365	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → (((coe1‘𝐴)‘(𝑥 − 𝐷)) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − (𝑥 − 𝐷)))) = (((coe1‘𝐴)‘(𝑥 − 𝐷)) × 𝐶))
104	86, 94, 103	3eqtrd 2768	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))))) = (((coe1‘𝐴)‘(𝑥 − 𝐷)) × 𝐶))
105	104	anassrs 467	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ 𝐷 ≤ 𝑥) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))))) = (((coe1‘𝐴)‘(𝑥 − 𝐷)) × 𝐶))
106	1	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → 𝑅 ∈ Ring)
107	3	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → 𝐶 ∈ 𝐾)
108	4	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → 𝐷 ∈ ℕ0)
109	54	ad2antll 729	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → (𝑥 − 𝑦) ∈ ℕ0)
110	54	nn0red 12446	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0...𝑥) → (𝑥 − 𝑦) ∈ ℝ)
111	110	ad2antll 729	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → (𝑥 − 𝑦) ∈ ℝ)
112	33	ad2antlr 727	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → 𝑥 ∈ ℝ)
113	35	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → 𝐷 ∈ ℝ)
114	47	ad2antll 729	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → 𝑦 ∈ ℕ0)
115	114	nn0ge0d 12448	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → 0 ≤ 𝑦)
116	47	nn0red 12446	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (0...𝑥) → 𝑦 ∈ ℝ)
117	116	ad2antll 729	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → 𝑦 ∈ ℝ)
118	112, 117	subge02d 11712	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → (0 ≤ 𝑦 ↔ (𝑥 − 𝑦) ≤ 𝑥))
119	115, 118	mpbid 232	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → (𝑥 − 𝑦) ≤ 𝑥)
120		simprl 770	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → ¬ 𝐷 ≤ 𝑥)
121	112, 113	ltnled 11263	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → (𝑥 < 𝐷 ↔ ¬ 𝐷 ≤ 𝑥))
122	120, 121	mpbird 257	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → 𝑥 < 𝐷)
123	111, 112, 113, 119, 122	lelttrd 11274	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → (𝑥 − 𝑦) < 𝐷)
124	111, 123	gtned 11251	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → 𝐷 ≠ (𝑥 − 𝑦))
125	20, 5, 6, 7, 8, 9, 10, 106, 107, 108, 109, 124	coe1tmfv2 22159	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦)) = 0 )
126	125	oveq2d 7365	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))) = (((coe1‘𝐴)‘𝑦) × 0 ))
127	45	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → (coe1‘𝐴):ℕ0⟶𝐾)
128	127, 114	ffvelcdmd 7019	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → ((coe1‘𝐴)‘𝑦) ∈ 𝐾)
129	106, 128, 81	syl2anc 584	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → (((coe1‘𝐴)‘𝑦) × 0 ) = 0 )
130	126, 129	eqtrd 2764	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))) = 0 )
131	130	anassrs 467	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ ¬ 𝐷 ≤ 𝑥) ∧ 𝑦 ∈ (0...𝑥)) → (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))) = 0 )
132	131	mpteq2dva 5185	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ ¬ 𝐷 ≤ 𝑥) → (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦)))) = (𝑦 ∈ (0...𝑥) ↦ 0 ))
133	132	oveq2d 7365	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ ¬ 𝐷 ≤ 𝑥) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))))) = (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ 0 )))
134	1, 22	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Mnd)
135	20	gsumz 18710	. . . . . . 7 ⊢ ((𝑅 ∈ Mnd ∧ (0...𝑥) ∈ V) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ 0 )) = 0 )
136	134, 24, 135	sylancl 586	. . . . . 6 ⊢ (𝜑 → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ 0 )) = 0 )
137	136	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ ¬ 𝐷 ≤ 𝑥) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ 0 )) = 0 )
138	133, 137	eqtrd 2764	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ ¬ 𝐷 ≤ 𝑥) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))))) = 0 )
139	18, 19, 105, 138	ifbothda 4515	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ0) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))))) = if(𝐷 ≤ 𝑥, (((coe1‘𝐴)‘(𝑥 − 𝐷)) × 𝐶), 0 ))
140	139	mpteq2dva 5185	. 2 ⊢ (𝜑 → (𝑥 ∈ ℕ0 ↦ (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦)))))) = (𝑥 ∈ ℕ0 ↦ if(𝐷 ≤ 𝑥, (((coe1‘𝐴)‘(𝑥 − 𝐷)) × 𝐶), 0 )))
141	17, 140	eqtrd 2764	1 ⊢ (𝜑 → (coe1‘(𝐴 ∙ (𝐶 · (𝐷 ↑ 𝑋)))) = (𝑥 ∈ ℕ0 ↦ if(𝐷 ≤ 𝑥, (((coe1‘𝐴)‘(𝑥 − 𝐷)) × 𝐶), 0 )))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  Vcvv 3436   ∖ cdif 3900  ifcif 4476  {csn 4577   class class class wbr 5092   ↦ cmpt 5173  ⟶wf 6478  ‘cfv 6482  (class class class)co 7349  ℂcc 11007  ℝcr 11008  0cc0 11009   < clt 11149   ≤ cle 11150   − cmin 11347  ℕ₀cn0 12384  ...cfz 13410  Basecbs 17120  ._rcmulr 17162   ·_𝑠 cvsca 17165  0_gc0g 17343   Σ_g cgsu 17344  Mndcmnd 18608  ._gcmg 18946  mulGrpcmgp 20025  Ringcrg 20118  var₁cv1 22058  Poly₁cpl1 22059  coe₁cco1 22060

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-iin 4944  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-isom 6491  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-of 7613  df-ofr 7614  df-om 7800  df-1st 7924  df-2nd 7925  df-supp 8094  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-2o 8389  df-er 8625  df-map 8755  df-pm 8756  df-ixp 8825  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-fsupp 9252  df-sup 9332  df-oi 9402  df-card 9835  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-nn 12129  df-2 12191  df-3 12192  df-4 12193  df-5 12194  df-6 12195  df-7 12196  df-8 12197  df-9 12198  df-n0 12385  df-z 12472  df-dec 12592  df-uz 12736  df-fz 13411  df-fzo 13558  df-seq 13909  df-hash 14238  df-struct 17058  df-sets 17075  df-slot 17093  df-ndx 17105  df-base 17121  df-ress 17142  df-plusg 17174  df-mulr 17175  df-sca 17177  df-vsca 17178  df-ip 17179  df-tset 17180  df-ple 17181  df-ds 17183  df-hom 17185  df-cco 17186  df-0g 17345  df-gsum 17346  df-prds 17351  df-pws 17353  df-mre 17488  df-mrc 17489  df-acs 17491  df-mgm 18514  df-sgrp 18593  df-mnd 18609  df-mhm 18657  df-submnd 18658  df-grp 18815  df-minusg 18816  df-sbg 18817  df-mulg 18947  df-subg 19002  df-ghm 19092  df-cntz 19196  df-cmn 19661  df-abl 19662  df-mgp 20026  df-rng 20038  df-ur 20067  df-ring 20120  df-subrng 20431  df-subrg 20455  df-lmod 20765  df-lss 20835  df-psr 21816  df-mvr 21817  df-mpl 21818  df-opsr 21820  df-psr1 22062  df-vr1 22063  df-ply1 22064  df-coe1 22065

This theorem is referenced by:  coe1tmmul2fv  22162  coe1sclmul2  22168